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MathsJEE Main 24 Jan 2026, Shift 2·Q4·MCQ·Moderate

Vectors · Projection of a vector

Let a⃗=2i^−j^−k^\vec a = 2\hat i - \hat j - \hat k, b⃗=i^+3j^−k^\vec b = \hat i + 3\hat j - \hat k and c⃗=2i^+j^+3k^\vec c = 2\hat i + \hat j + 3\hat k. Let v⃗=a⃗+λb⃗\vec v = \vec a + \lambda\vec b, λ∈R\lambda \in \mathbb{R}, be the vector in the plane of the vectors a⃗\vec a and b⃗\vec b, such that the length of its projection on the vector c⃗\vec c is 114\dfrac{1}{\sqrt{14}}. Then ∣v⃗∣\lvert\vec v\rvert is equal to
  1. (1)212\dfrac{\sqrt{21}}{2}
  2. (2)13
  3. (3)352\dfrac{\sqrt{35}}{2}
  4. (4)7
We corrected a printing error in the original paper

Assumption “v⃗=a⃗+λb⃗\vec v = \vec a + \lambda\vec b, λ∈R\lambda \in \mathbb{R}” added; the source states only that v⃗\vec v lies in the plane of a⃗\vec a and b⃗\vec b, which leaves ∣v⃗∣\lvert\vec v\rvert undetermined (the source prints “Ans. (Bonus)” and “NTA (3)”); key (3) holds when the coefficient of a⃗\vec a is 1.

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